Relational-Topological Geometry of Couplings Stabilizing Quantum States: the KQM log Estimator, Entanglement and Decoherence within the USC Framework.

Introduction This article presents the quantum module of the USC formalism, namely the Universal Structural Code, developed as a classificatory layer within the PJM/LOM programme and the GTSFC–USC–GTCW framework. The work does not modify quantum mechanics, the Schrödinger equation, the density matrix formalism, Lindblad dynamics, or the standard theory of decoherence. It presents an additional operational description of quantum state stability, based on a relational and topological analysis of phase coherence, entanglement, purity, decoherence, spin chiral structure, and topological protection. The central element of the article is the estimator Klog⁡QMK^QM_KlogQM, which aggregates selected observables of quantum mechanics into a single scale of relational non closure. In this formulation, a quantum state is classified not only through energy, entropy, purity, or a single entanglement measure, but through the multichannel consistency of its relational structure. Quantum stability is therefore described as the simultaneous preservation of phase coherence, topological robustness, entanglement structure, spin correlations, and a low level of decoherence. The significance of the work for established physics lies in the ordering of standard quantum mechanical observables within a new diagnostic language. The estimator Klog⁡QMK^QM_KlogQM can be treated as a cost function or classification map acting on density matrices ρ∈D (H) D (H) ρ∈D (H). In its developed form, the formalism incorporates the geometry of quantum states, including fidelity, the Bures distance, the Fubini Study metric in the pure state limit, and the covariance aware form of channel mismatch. In this way, the module is embedded in the known tools of quantum information physics and does not introduce a competing dynamical apparatus. The significance of the work for quantum computer architecture is related to the problem of maintaining coherence, entanglement, noise robustness, and logical stability of quantum states over time. In practical quantum architectures, the mere preparation of a qubit or an entangled state is not sufficient for useful information processing. The essential issue remains the ability of a system to preserve the relational structure of the state under noise, dephasing, environmental coupling, and local errors. The estimator Klog⁡QMK^QM_KlogQM is presented here as a candidate multichannel indicator of quantum state degradation, applicable to the comparison of Bell states, decohered mixtures, topologically protected systems, the Kitaev chain, the toric code, and future fault tolerant architectures. Two levels of demonstration are presented. The first concerns a Bell state subjected to dephasing, where the estimator increases monotonically with the loss of phase coherence, entanglement, and purity. The second concerns a minimal topological test based on the Kitaev chain, where the topologically protected phase obtains a lower value of Klog⁡QMK^QM_KlogQM than the trivial phase. These results have the status of an operational demonstration, not a full ontological validation of USC. Their significance lies in showing that the quantum module of USC can be formulated in a language consistent with established quantum mechanics, state geometry, and quantum information theory. The main contribution of the article is the indication that quantum state stability can be analysed not only through individual measures, such as purity, von Neumann entropy, concurrence, or fidelity, but through their relational structure. In this sense, the article proposes a transition from the classification of individual properties of a state to the classification of its multichannel coherence. The question of whether a state is pure, entangled, or topological is supplemented by the question of whether the set of its properties forms a stable, relationally closed quantum configuration. The scientific status of the work is defined as QM-PROGRAM. This means that the presented formalism has the character of a mathematically coherent and testable classification programme, consistent with established physics, while still requiring further numerical validation. The next stage is defined as a technical annex containing full mathematical and physical derivations, simulations of open quantum systems, Lindblad tests, analysis of the noisy Kitaev chain, the toric code, null models, ROC/AUC curves, and comparison of the Klog⁡QMK^QM_KlogQM estimator with individual standard measures. In a broader sense, the article shows that USC can be formulated as a tool for classifying and testing the stability of quantum structures. This form of the formalism creates a bridge between the geometry of quantum states, quantum information theory, decoherence, topological protection of states, and practical quantum computer engineering. If further numerical tests confirm the advantage of the estimator over null models, the Klog⁡QMK^QM_KlogQM module may acquire the status of a useful diagnostic tool for analysing the stability of quantum states and quantum computing architectures. Keywords Universal Structural Code, USC, PJM, LOM, GTSFC, quantum mechanics, geometry of quantum states, decoherence, entanglement, fidelity, Bures distance, density matrix, Kitaev chain, topological protection of states, quantum computers, Klog⁡QMK^QM_KlogQM, relational closure, quantum information, open quantum systems.